Screw thread



Sept. 1-3, 1932'.

N. TRBOJEVICH SCREW THREAD Filed March 2, 1931 2 Sheets-Sheet 1 IINVENTOR A t/ 02 77'50/ era's/3 ATTORNEYS p 3 N. TRBOJEVICH 1,876,796

SCREW THREAD Filed March 2. 1931 2 Sheets-Sheet 2 W ATTORNEY;

Patented Sept. 13, 1932 UNITED STATES NIKOLA TRBOJ'EVICH, F DETROIT,MICHIGAN SCREW THREAD Application filed March 2, 1931. Serial No.519,578.

The invention relates to a novel form of screw thread applicable toworms, screws. and nuts.

The invention resides in so forming the thread surfaces in a screw thatthey will possess a negative Gaussian or hyperbolic curvature at anypoint thereof. This is accomplished by forming the contours curved andconcave in the axial plane,.and convex in the plane perpendicular to thesaid axis, the opposite being true for the cooperating nuts.

Heretofore, the screw surfaces were convex at every point thuspossessing only a single family or system of the so-called asymptoticlines, i. e. lines alon which the radii normal to the surface are innite. Ihave discovered that for the work of this character hyperbolic orsaddle-shaped surfaces are greatly to be preferred inasmuch as theypossess two families of asymptotic lines forming a network in thesurface.

The object of this invention is to increase the number and densityof'the asymptotic lines in a screw surface and thereby stifien the same.

Another object is to form the cooperating contours in the screw and nutof a slightly different curvature, thus providing a point contact onlywhen not loaded and a surface contact when under load. By thisarrangement I'preserve the tangential nature of contact in spite ofcertain (limited) manufacturing inaccuracies.

In a modification of this principle a thread surface is constructedwhich has a constant efficiency factor at any point thereof.

In another modification I so select the principal radii of curvaturethat the asymptotic lines will form an orthogonal network therebyproducing the maximum surface rigidity or stiffness.

In the drawings Figure 1 is the elevation of my improved screw and nut;

Figure 1a is an enlarged view of the cooper-- ating thread surfaces withthe curvature exaggerated;

Figure 2 is the Dupins indicatrix taken at the point A, Figure 1;

Figures 3 to 7 are diagrams explanatory of the structural advantagespossessed by hyperbolic surfaces in general;-

Figures 8 to 11 are geometrical diagrams from which the equations 3 to20 are deduced;

Figure 12 shows a portion of a minimal surface which may be constructedon this principle. 1

In Figure 1 the screw 11 has a helical thread 12 wound about thecylinder axis 13. The flanks 1 1 of the screw and the flanks 15 of thenut 16 are the circular arcs 17 and 18 respectively struck from thecenters 0 and O and tangent at the pitch point A. The screw radius A Ois by about ten per cent longer than the nut radius A 0, saiddiscrepancy in length being selected with a view of afi'ording asuitable manufacturing tolerance without sacrificing the tangentialnature of contact essential for a smooth and dependable operation.

A-bottom clearance 19 is provided at the roots of the cooperatingthreads. The nominal pressure angle ,8; may be" arbitrarily selected andits value will be about 30 for ordinary screws and from 14% to 20 forwork screws. g

The principal radii of curvature in the new thread surface are AB =R andA0 =R Figure 1, the influence of the usually small 'helix angle a beingneglected. The first radius R represents, therefore, the curvature alongthe helix and is positive for the screw and negative for the nut while Rpertains to the axial contours and is always of the opposite sign to RIt is now possible to determine the radius of curvature R of the surfaceat any desired normal plane passing through the normal to the surface atthe point A and forming a known angle with either one of the principalplanes R and R By calculation, this is done by solving the Eulerequation and, graphically, the radius may be scaled ofl from the Dupinindicatrix, Figure 2.

With the half axes /1? and /13 respectively two pairs of conjugatehyperbolas are 10'? constructed. In any selected direction the radius ofcurvature will be /TZ=AD Of particular interest -are the radii in theasymptotes 24 and 25 as they are both infinite. The asymptotic angle 8is readily found from the equation merge one into the other with aninflexion' along the asymptotes 24 and 25 and it will be obvious evenwithout a mathematical calculation that the radii of curvature at thepoints of junction must be infinite as the surface is neither convex orconcave at any such point.

The nature of the surface in the neighborhood of the point A maybe-further studied by following the closed orbit 26 in the direction ofthe arrow. Starting from the plus X axis the path is downhill until theplus Y axis is reached after which the path turns uphill toward theminus X axis and so on. In a complete cycle two maxima, two minima andfour points of inflexion are passed as it will be further understood bystudyin the upper half of the Figure 3 which shows t 1e development ofthe orbit 26 in the plane of paper. The tangent plane T crosses thesurface with an inflexion at the asymptotes as shown in Figure 6. Thenetwork of the asymptotic lines comprising two systems 27 and 28 isshown in Figure 5. i

On the other hand, in the standard U. S.

thread 29 (Figure 4:) there is only a single system of the asymptoticlines 30,;Zand the orbit 2,611) as shown in the lower of the Figure 3crosses only twice in a cycle the locus 31 of infinite radius ofcurvature, see also Figure 7.

I shall now show the solution of the problem in which the radius R is soselected with reference to the pitch radius g of the screw that theresulting hyperbolic surface will have a constant efficiency at anypoint thereof. The practical significance of. this improyement lies inthe fact that while in the ordinary screw the efliciency is variablefrom v the roots to the tips of the thread due to a form lack ofefficiency throughout the surface a variable helix angle, in myconstruction I counteract the said variation byalso varying the pressureangle at the same time. A constant efliciency also means a constant oruniand by this method screws of an increased frictional holding powermay be constructed.

In igu re 8 let the screw element 1 be subjected to a vertical force Qrepresenting the load uponthe nut; If the friction f and the pressureangle ,8 of the screw thread be disregarded for the time being, an idealforce P will be sufficient to hold the force Q in balance, namely P Qsin a. (3)

Due to the existence of the pressure angle and friction the value of Pwill be increased to P, viz;

where 71. is the lead of the screw, y the (variable) radius. Thededuction of the Equation (4) may be found in. Freytag, Maschinenbau,

1920 edition page 136. The efficiency of the screw will then be and thelack of efficiency F=Z1 after suitable simplifications will be aw-am.tan a cos 6+) (6) where h tan a= 7 The helix angle or decreases as theradius 7' increases, see Figure 10. The value of tan 0: in the numeratorof the Equation (6) is readily negligible for small helix angles. Thus,

f tanfa cos B +f In order to keep the value of F a constant it isnecessary that tan 01 cos ,B=const.=tan .1. cos 3., (8)

where the null suflix denotes quantities pertaining to the pitch pointA, Figure 9.

From Figure 10 let where h is the lead per radian. Then sin c= sinc (15) I The latter, however, is

From (8) and (12) cos B= -c0s B G- cos 6., (16) '0 '0 Squaring andadding the last two equations 2 1=' 2 o+ +2' .3 sin [5 p2 sm 5 p As thedistance a: is relatively small, its square may be omitted.

0=2 sin so g-00's 3.? (18) from which 00 vanishes.

sin 6 p r cos 260 Q. E. D. (19) For example, let B =3O. Then sin= andcos34 Z 293121: tan 6 J I s 50 :1: tan 6., (21) showing that the asymptoticlines are symmetrically disposed relative to the pitch helix andintersect the same at an angle B The tooth curve is necessarily concavei the screw because if tan 0: cos B=constant then also 1 sec B must be aconstant because of (10) an equilateral hyperbola in r and sec ,3,Figure 11. Thus, when 1- increases sec B (and also B) must diminishproving that the pressure angle is greater at the roots than at thetips. In other words the curve is such as shown in Figure 9.

If it is desired to construct a surface of this kind possessing thegreatest possible surface rigidity, the concave radius p should be soselected that the average radius of curvature of the surface be themaximum. It is possible to do so. In fact, it is possible to make theaverage radius equal to infinite. According to Cauchy where R is themean or average radius. If

a load to transmit.

'a straight line, p equals infinite and from 23 friction screw, Equation19 this be infinite, the left hand side of the equation will become zeroand I have a 145 Rule $2.--If I select the radius of the tooth flankequal to the pitch diameter (for 30 degrees pressure angle) the surfacewill have the maximum strength and the asymptotic lines will cross eachother at right angles.

In action the new screw and nut will only contact with a point contactbecause I employ a longer radius in the screw flank than in the nutflank. Thus the screw or the nut is easily manipulated when not loadedand may be constructed with less backlash than the present screws. Whenloaded, the mating concave and convex curves Will flatten out andproduce a b aring area. It will be seen that this principle is soundbecause a bearing area is not needed unless there is At the same timethe mean radius of curvature is enormously increased. As was stated inconnection with the Equation 23 I can make the mean radius of thesurface of any magnitude whatever by suitably selecting the flank radiusp. On the other hand, in common screws'thefiank is '0 sin 3 (26) i. e.the mean surface radius cannot be greater than the pitch diameter of thescrew, for 30 degrees pressure angle.

'The flank radius 0A= of the constantis readily obtained by a simpleprojection as shown in Figure 1. I drop theperpendicular A C upon thenormal A B thus obtaining the point C in the axis 13 of the screw.Another perpendicular C O is dropped (this line upon the axis 13) thusfinding the center of curvature O in the normal B A O. This'satisfiesthe Equation 19 perfectly because cos 6.,

as measured from the roots toward the tips of the thread.

2. A screw having a helical thread of a constant pitch wound about anaxis in which the thread contours in the axial plane are substantiallycircular arcs drawn from centers lying on the outside of the pitchcylinder in such a manner that the said contours are concave on bothsides of the thread in the axial plane thereof and have a continuouslyvarying pressure angle which diminishes from the roots toward the tipsof the thread.

3. A screw having a helical thread of a constant pitch wound about apitch cylinder and concave curved flanks at its both sides measured inthe axial planet-hereof, in which the radius of curvature of the saidflanks is selected with reference to the pitch diameter and pressureangle in such a manner that the asymptotic lines will cross the pitchhelix at an angle substantially equal to the said pressure angle therebyobtaining a screw of an increased self locking ability.

4. A screw having a cylindrical body and a helical thread wound thereonin which the thread cross contours in the axial plane are curved andconcave and in which the radius of curvature of the said contours issubstantially equal to the normal surface radius as measured in theplane tangent to pitch helix thereby producing an orthogonal network ofasymptotic lines and a surface of an increased load bearing ability. 7,

5. A cooperating screw and nut in which the tliread contours in theaxial plane are curved, concave and possess an ever diminishing pressureangle from the roots toward the tips of the thread, in the screw, andare curved, convex and tangential at a point to the mating screwcontours in the nut.

6. A cooperating screw and nut in which the contours of the screw threadin the axial plane are curved and concave at all points thereof and themating contours in the nut are curved and convex, the relation beingsuch that the mating contours in the members are tangent to each otherand the radius of curvature of the concave contours is greater than theradius of the convex contours.

7. A screw thread formed about an axis in a helix 1n such a manner thatthe radius of curvature of its contours in the axial plane lies at oneside of the thread surface and the radius of the contours in the planeperpendicular to the axis lies at the other side of the said surfacethus forming a saddle shaped surface having two series of asymptoticlines. In testimony whereof I affix my signature.

. NIKOLA TRBOJEVICH.

